Grid Curves on Parametric Surfaces

Grid Curves on Parametric Surfaces

$u equals a u = a$
$v equals b v = b$

x equals sine of u times cosine of v x = sin u cos v

y equals sine of u times sine of v y = sin u sin v

z equals cosine of u z = cos u

CONCEPT

You will learn about:

Grid curves within parametric surfaces.

pile of disassembled drill screws for getting ground water

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Many machine parts, such as augers, are shaped similarly to corkscrews. Such shapes can be modeled with parametric surfaces. A vector function vector function r of u and v

\mathbf{r}\left( u,v \right)

of two parameters u

u

and v

v

can be defined by vector function r of u and v equals x of u and v multiplied by ort i plus y of u and v multiplied by ort j plus z of x and y multiplied by ort k

\mathbf{r}\left( u,v \right) = x\left( u,v \right)\mathbf{i}+ y\left( u,v \right)\mathbf{j} + z\left( u,v \right)\mathbf{k}

for a region capital D

D

in the u and v

\text{uv}

-plane. If u

u

is kept constant by putting u equals u zero

u = u_{0}

, then vector function r of u zero and v

\mathbf{r}\left( u_{0},v \right)

becomes a vector function of the single parameter v

v

and defines a curve lying on the parametric surface. Similarly, if v

v

is kept constant by putting v equals v zero

v = v_{0}

, then vector function r of u and v zero

\mathbf{r}\left( u,v_{0} \right)

becomes a vector function of the single parameter u

u

and defines another curve lying on the surface.

WHEN WOULD I USE THIS

Grid curves can be used to visualize the surface formed by a vector function of two parameters. They represent families of space curves that can be used to construct the surfaces.

Blue pump compressor for washing cars, indoor. Cleaning concept.

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Repair technicians must regularly replace parts on machines, such as compressors. Such compressors often use spiral tubes that can be modeled using parametric surfaces. Grid curves in one variable describe a helix, while grid curves in another variable describe circles.

Abstract polygonal space low poly dark background with connecting dots and lines. Connection structure.

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Graphic designers use computers to render many surfaces. Grid curves are used by many computer rendering systems to approximate surfaces using sets of curves defined by pairs of values in the parameter plane.

Globe isolated on white

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Cartographers can model the planet Earth using the parametric surface representation of a sphere. In this representation, grid curves in one variable are lines of longitude that connect the North Pole and the South Pole. Grid curves in the other variable are latitude lines.

INSTRUCTIONS

Select one of the four surfaces from the pull-down menu. Parametric equations for the surface are shown at the right. Use the mouse to drag the red vertical line, $u equals a u = a$ , or the blue horizontal line, $v equals b v = b$ , in the $u v u v$ -plane. The corresponding grid curve on the surface will move accordingly. You may wish to rotate the axes of the 3D graph with the mouse for different views of the surfac and its grid curves.

For each surface, how would you describe the family of grid curves with $u u$ constant? With $v v$ constant?

EXERCISES